Representation homology, Lie algebra cohomology and the derived Harish-Chandra homomorphism

Yuri Berest ; Giovanni Felder ; Sasha Patotski ; Ajay C. Ramadoss ; Thomas Willwacher

Journal of the European Mathematical Society, Tome 19 (2017) no. 9, pp. 2811-2893.

Résumé

We study the derived representation scheme DRepn​(A) parametrizing the n-dimensional representations of an associative algebra A over a field of characteristic zero. We show that the homology of DRepn​(A) is isomorphic to the Chevalley–Eilenberg homology of the current Lie coalgebra gln∗​(Cˉ) defined over a Koszul dual coalgebra of A. This gives a conceptual explanation to some of the main results of [BKR] and [BR], relating them (via Koszul duality) to classical theorems on (co)homology of current Lie algebras gln​(A) . We extend the above isomorphism to representation schemes of Lie algebras: for a finite-dimensional reductive Lie algebra g, we define the derived affine scheme DRepg​(a) parametrizing the representations (in g) of a Lie algebra a; we show that the homology of DRep_g(a) is isomorphic to the Chevalley–Eilenberg homology of the Lie coalgebra g∗(Cˉ), where C is a cocommutative DG coalgebra Koszul dual to the Lie algebra a. We construct a canonical DG algebra map Φg​(a):DRepg​(a)G→DReph​(a)W , relating the G-invariant part of representation homology of a Lie algebra a in g to the W-invariant part of representation homology of a in a Cartan subalgebra of g. We call this map the derived Harish-Chandra homomorphism as it is a natural homological extension of the classical Harish-Chandra restriction map.

Détail

DOI : 10.4171/jems/729

Classification : 17-XX, 16-XX, 18-XX, 53-XX
Keywords: Derived representation scheme, Lie algebra cohomology, Chevalley–Eilenberg complex, Harish-Chandra homomorphism, Koszul duality, Macdonald identity